My mathematical work is supported by NSF grant DMS-1100784.
I am an editor for
Journal of the AMS,
Algebra and Number Theory, and
IMRN. Please consider submitting appropriate papers to these journals. All
such submissions must go through the journal website; papers cannot be submitted
through editors, and any submissions on famous open problems are subject to the strict rules as described
at the lower half of this page.
[JAMS has acceptance standards on par with Annals of Math and accepts around 30 papers
per year across all areas of math, ANT is the top journal for specialized papers in algebra and number theory, and IMRN is
a general-interest journal with an acceptance standard roughly at the level just below that of Duke Math Journal.]
If you are taking a class from me then you should find a functioning
link to it below (with course information, homeworks, and so on).
If several days go by without a response to email,
I am probably away from home.
Number theory and representation theory seminar
Analytic number theory, algebraic number theory,
arithmetic geometry, automorphic forms, and even some things
not beginning with the letter "a". It's a big subject.
Schedule and notes for the 2016-17 Seminaire Deligne-Laumon
Schedule and notes for the 2015-16 Seminaire BSD/Bloch-Kato
Schedule and notes for the 2014-15 Seminaire Scholze
Schedule and notes for the 2013-14 Seminaire Jacquet-Langlands
Schedule and notes for the 2012-13 Seminaire Shimura
Schedule and notes for the 2011-12 Seminaire Darmon
Schedule and notes for the 2010-11 Mordell seminar
Schedule & notes for the 2009-10 modularity lifting seminar
Links to current courses:
Undergraduate algebraic number theory
Here are handouts and homeworks from some past undergraduate courses:
Undergraduate algebraic geometry ,
Galois theory
Here are handouts and homeworks from some past graduate courses:
Linear algebraic groups I ,
Linear algebraic groups II ,
Abelian varieties,
Alterations ,
Graduate Algebraic Number Theory,
Class field theory,
Modular curves,
Compact Lie groups ,
Algebraic Geometry intensive reading course (fall) ,
Algebraic Geometry intensive reading course (winter) ,
Intro Graduate Algebra (winter)
Some differential geometry
I once taught an introductory differential geometry course
and was rather disappointed with the course text,
so I went overboard (or crazy?) and wrote several hundred pages of stuff to supplement
the book.
If you are learning elementary differential geometry, maybe you'll find
some of these handouts to be interesting. Most likely I will never again
teach such a course.
Ross Program,
PROMYS Program,
and
Epsilon Fund
Some mathematicians are entirely self-created;
for the rest of us, assistance and encouragement
early on is helpful. These programs do an excellent job in that direction.
Research
First, some caveats.
0. Links to files undergoing revision may be temporarily disabled.
1. If you want to know where something below was published
(if it has appeared in print), then please look on MathSciNet.
2. Someday I should join the 21st century and post
papers on the arxiv, at least after I can no longer make changes
to the version to be published. In particular, I should post
my old papers that have already appeared. Unfortunately (?), I tend
to rewrite things too many times and don't wish to keep
posting revision on top of revision on the arxiv. Posting things
here (even in far-from-final state) partially compensates for my
pedantry, I hope.
Are you looking for how to get a copy of
the pseudo-reductive book with Gabber and Prasad? Or a draft copy of the CM book with Chai and Oort?
If so, scroll down to the "Book" section below.
Classification of pseudo-reductive groups
pdf
Reductive group schemes (notes for "SGA3 summer school").
pdf
This proves main results from SGA3 using Artin's work and the dynamic method from "Pseudo-reductive groups" to simplify proofs.
Here is a link
to some other notes from the summer school, inspired by a lecture given there by
B. Gross on non-split groups over the integers, and a related file by J.-K. Yu providing
computer code used in some of the computations therein.
Algebraic independence of periods and logarithms of Drinfeld modules (by C-Y. Chang, M. Papanikolas).
pdf
I provided the appendix (a pseudo-application of pseudo-reductive groups).
Lifting global representations with local properties
pdf
Universal property of non-archimedean analytification.
pdf
Descent for non-archimedean analytic spaces (with M. Temkin).
pdf
Finiteness theorems for algebraic groups over function fields.
pdf
Moishezon spaces in rigid geometry.
pdf
Nagata compactification for algebraic spaces (with M. Lieblich, M. Olsson).
pdf
Arithmetic properties of the Shimura-Shintani-Waldspurger correspondence
(by K. Prasanna).
pdf
I provided the appendix.
Non-archimedean analytification of algebraic spaces (with M. Temkin).
pdf
Chow's K/k-image and K/k-trace, and the Lang-Neron theorem
(via schemes).
pdf
This largely expository note improves the non-effective classical version of
the Chow regularity theorem, and generally uses infinitesimal methods and
flat descent to replace Weil-style proofs that I could not understand.
It is also cited in "Root numbers and ranks in positive characteristic" below.
Prime specialization in higher genus II (with K. Conrad and R. Gross).
pdf
Prime specialization in higher genus I (with K. Conrad).
pdf
Higher-level canonical subgroups in abelian varieties.
pdf
Modular curves and rigid-analytic spaces.
pdf
Relative ampleness in rigid-analytic geometry.
pdf
Root numbers and ranks in positive characteristic
(with K. Conrad and H. Helfgott).
pdf
Arithmetic moduli of generalized elliptic curves.
pdf
Edixhoven has a different approach to these matters
when the moduli stacks are Deligne-Mumford.
Prime specialization in genus 0 (with K. Conrad and R. Gross).
pdf
Modular curves and Ramanujan's continued fraction (with B. Cais).
pdf
A short erratum to this paper. pdf
"On quasi-reductive group schemes"
(by G. Prasad and J-K. Yu).
pdf
I provided the appendix.
The Möbius function and the residue theorem (with K. Conrad).
pdf
This is a companion to "Prime specialization in genus 0"
above.
J_{1}(p) has connected fibers
(with S. Edixhoven and W. Stein).
pdf
Finite-order automorphisms of a certain torus.
pdf
Gross-Zagier revisited (with appendix
by W. R. Mann).
pdf
Power laws for monkeys typing randomly: the case of unequal
probabilities (with M. Mitzenmacher).
pdf
Approximation of versal deformations (with A. J. de Jong).
pdf
A modern proof of Chevalley's theorem on algebraic groups.
pdf
Component groups of purely toric quotients
(with W. Stein).
pdf
On the modularity of elliptic curves over Q
(with C. Breuil, F. Diamond, R. Taylor).
pdf
Inertia groups and fibers.
pdf
Correction to "Inertia groups and fibers"
pdf
Irreducible components of rigid spaces.
pdf
Modularity of certain potentially Barsotti-Tate
representations (with F. Diamond and R. Taylor).
pdf
Remarks on mod-l^{n} representations with
l = 3, 5
(with S. Wong).
pdf
Ramified deformation problems.
pdf
Finite group schemes over bases with low ramification.
pdf
Books
Here are some books, pre-books, etc.
Complex multiplication and lifting problems (with C-L. Chai, F. Oort)
order while supplies last!
(this link might be behind a firewall; the ISBN number is ISBN-10: 1-4704-1014-1)
Pseudo-reductive groups (with O. Gabber, G. Prasad)
2nd edition!
Grothendieck duality and base change
tar
Clarifications and corrections to "Grothendieck duality and base change"
pdf
An addendum to Chapter 5 of "Grothendieck duality and base change"
pdf
My book on Galois representations and modular forms is
still undergoing revisions. (It
is now shorter than it was before, with much better proofs;
if you have an earlier version, please burn it.) The
following link has been disabled.
Modular forms and the Ramanujan conjecture
pdf
Here is a very rough draft of a book with M. Lieblich (more to be
included, such as exercises, applications to the theory of
the monodromy pairing and the p-adic good reduction criterion, etc., and much to
be revised). Whoops, the link is now disabled (due to
some strange errors that need to be fixed).
Galois representations arising from p-divisible groups
pdf
Notes
Here are some expository things I have written.
These may be updated without warning. Links to files undergoing
revision may be temporarily disabled.
The basic guiding principles for deciding what to write up and post here
are two-fold.
1. Interesting (to me?) alternative proofs
of known results, or explanations of
important topics that can be difficult for beginners
to learn are fair game.
(As the literature improves, I may augment postings
accordingly.)
2. With all due respect to
the role of Andre Weil in the development of
algebraic geometry, nobody should ever again have to
read Weil's "Foundations of algebraic geometry": EGA must be an adequate
logical starting point for the subject. Hence, if there is
an important, interesting, or useful theorem
whose published proofs use pre-Grothendieck
methods in such an essential way so as to
render them impenetrable to later generations
(or to me?),
and if I have a need to understand
why the theorem is true and consequently I figure out a scheme-theoretic
proof, then I'll try to write it up.
The structure of solvable groups over general fields. pdf
These are notes that mildly revise Appendix B of the "pseudo-reductive" book, to accompany some lectures at the 2011 "SGA3 summer school" in Lumimy.
Some notes on topologizing the adelic points of schemes,
unifying the viewpoints of Grothendieck and Weil.
pdf
Some lecture notes on p-adic Hodge theory, from a course
I taught with Olivier Brinon at the 2009 CMI summer school on Galois
representations. It is undergoing regular revision; not yet in final form (so corrections
welcome).
pdf
2007 Arizona Winter School lectures on rigid geometry
pdf
This appeared as a chapter in a book produced by the Winter School.
Formal GAGA for Artin stacks.
pdf
The paper "On proper coverings of Artin stacks" by M. Olsson gives
a different point of view on this topic.
Rosenlicht's unit theorem
pdf
Rosenlicht proved an extremely interesting theorem on
the structure of units on a product of varieties. Since
Rosenlicht's proof doesn't address the generality cited
without proof in SGA7, and it was written in archaic terminology,
this short note gives a "modern" treatment (and eliminates
the assumption on existence of rational points).
Integration in elementary terms
pdf
Have you told your calculus class that the Gaussian integral cannot
be computed in elementary terms, or likewise for elliptic integrals
when teaching a course on Riemann surfaces, but you personally have no idea how
such a (useless but pretty) result is proved? If so, then you
should definitely read Rosenlicht's article "Integration in finite terms"
in volume 79 of
the American Mathematical Monthly (in 1972). But if you have to explain
the proof to talented high school students, this short note (based on a talk
I gave to such students at CMI) may be helpful.
Keel-Mori theorem via stacks.
pdf
In this note, we use stacks instead of groupoids to give
a streamlined proof of the Keel-Mori existence theorem
for coarse moduli spaces (useless bonus: noetherian hypotheses eliminated).
Minimal models for elliptic curves.
pdf
The book "Algebraic geometry and arithmetic curves"
by Q. Liu treats much (but not all) of this material.
Cohomological descent.
pdf
WARNING: these notes are essentially written
in the context of topology and
etale sites; this is adequate for arithmetic
applications (potential semistability of l-adic
cohomology via alterations) but is woefully inadequate
for geometric applications (quasi-coherent cohomology on Artin stacks).
The level of generality
(and aspects of the writing style)
will be much-improved in the next version
so that the notes suffice to treat geometric applications too.
Descent for coherent sheaves on rigid-analytic spaces.
pdf
The paper "Coherent modules and their descent on relative
rigid spaces" by S. Bosch and U. Görtz gives a different approach
to this topic (pre-dating mine, as I found out later).
Nagata's compactification theorem (via schemes).
pdf
This is a detailed exposition of some
private notes of Deligne.
The paper "On compactification of schemes" by W. Lütkebohmert
gives a different approach to this topic in the noetherian case,
but Deligne's approach gives some striking results of general
interest for rational maps.
If you can understand what is going on in
Nagata's original paper then you are either very old
or very smart (or both!). These notes were eventually published
in Journal of the Ramanujan Math Society (vol. 22, 2007).
In the published version there was a tiny mistake in
one definition in the middle of a long proof, but this error
was localized and easy to fix. There was also a tiny error
in the appendix (never used in the main text). Here is the official erratum (but the
above link is to a file in which the errors are corrected).
pdf
The classical Riemann-Hilbert
correspondence.
pdf
Some exercises on group schemes and p-divisible groups.
Homework 1: pdf.
Homework 2: pdf.
Homework 3: pdf.
Homework 4: pdf.
These are the "homework" exercises for a week-long
instructional workshop for graduate students
co-organized with Andreatta and Schoof in May, 2005.
These were too many exercises for the amount of time given.
But if you have more than a week to spend on them then
perhaps some of the exercises will be helpful or interesting if you
are taking your first steps in this direction. Since
the lectures that naturally accompany these exercises
are not recorded here, a recommended substitute is some of the
written lecture notes from the ``Notes on complex multiplication''
(see above) and a lot of asparagus.
What do these 5 people have in common?
Draft of Andreatta's notes for course at 2009 CMI summer school
pdf